# Can you add two irrational numbers to get a rational number?

###### Wiki User

###### July 14, 2015 5:54PM

Yes Yes, the sum of two irrational numbers can be rational. A
simple example is adding sqrt{2} and -sqrt{2}, both of which are
irrational and sum to give the rational number 0. In fact, any
rational number can be written as the sum of two irrational numbers
in an infinite number of ways. Another example would be the sum of
the following irrational quantities [2 + sqrt(2)] and [2 -
sqrt(2)]. Both quantities are positive and irrational and yield a
rational sum. (Four in this case.) The statement that there are an
infinite number of ways of writing any rational number as the sum
of two irrational numbers is true. The reason is as follows: If two
numbers sum to a rational number then either both numbers are
rational or both numbers are irrational. (The proof of this by
contradiction is trivial.) Thus, given a rational number, *r*,
then for ANY irrational number, *i*, the irrational pair
(*i, r-i*) sum to *r*. So, the statement can actually be
strengthened to say that there are an infinite number of ways of
writing a rational number as the sum of two irrational numbers.